Such decomposition not only increases the dimensions of the original data but also supplies us with the extracted anomalies.
As shown in Figure REF , we begin the decomposition by approximating the trend line \(\mathbf {t}^{(k)}\)  with the locally weighted scatterplot smoothing (i.e., LOESS  [1]}).
Then, we divide the original data \(\mathbf {x}^{(k)}\)  by the approximated trend line to derive the detrended time seriesWe use the log transform of \(\mathbf {x}^{(k)}\)  to handle the situation that specific values of original data are zero..
We then partition the detrended time series into periods of cyclic sub-series where the cycle size is determined by the time interval of the dataset. As an example, the cycle size for a monthly dataset would be 12 (one year as a cycle). Then, we obtain the seasonal component (\(\mathbf {s}^{(k)}\) ) by grouping the detrended series in each period and deriving the average value of each period across the time series. Subsequently, the residual component (\(\mathbf {r}^{(k)}\) ) is derived by dividing the seasonal and trend segments from the original series.
